Control of Inflation: a singular stochastic control problem

A two-dimensional singular stochastic control problem with an infinite horizon, arising when the Central Bank tries to contain the inflation by acting on the nominal interest rate, is studied. It is shown that the problem admits a variational formulation which can be differentiated to lead to a stochastic differential game between the conservative and the expansionist tendencies of the Bank. This result also holds when a finite horizon is used in the model. For the infinite horizon case, substantial regularity of the free boundary associated to the differential game is obtained. Existence of an optimal policy is established and it is shown that the optimal process is a diffusion reflected at the boundary. Numerical results obtained by fitting the model to Canadian data of the past 15 years are given.